Neutron spectrum inversion method based on maximum likelihood expectation maximization theory

Abstract

The application provides a neutron spectrum inversion method based on a maximum likelihood expectation maximization theory, and the method comprises the following steps: S1, screening at least one group of detection foils; S2, placing the detection foils into a neutron field to be detected for irradiation, and measuring the radioactivity of the detection foils; and S3, obtaining a neutron spectrum by using a maximum likelihood expectation maximization inversion algorithm. The application adopts a likelihood function based on a Poisson distribution to derive an iterative equation, which is consistent with the statistical characteristics of a detection system, and the inversion result is more in line with physical reality. The application can guarantee that the spectrum is non-negative, does not need to be manually corrected, and has stronger physical rationality. The application does not need to directly solve a matrix inverse, has good ill-condition resistance, and has higher stability.

Description

Technical Field

[0001] This invention relates to the field of nuclear power, and in particular to a neutron energy spectrum inversion method based on the maximum likelihood expectation maximization theory. Background Technology

[0002] In existing technologies, the neutron energy spectrum is a key parameter describing the distribution of neutrons in different energy ranges within a neutron field, and it has wide applications in nuclear reactor physics, radiation protection, and accelerator-driven subcritical systems. Since neutrons are uncharged and their energy cannot be measured by direct ionization like charged particles, detection must rely on secondary effects produced when neutrons interact with atomic nuclei.

[0003] Currently, based on different physical principles, the methods for measuring neutron energy spectra can be mainly divided into the following seven categories: 1. Measure the energy of the recoil nucleus to obtain the characteristics of the scattered neutron; II. Measuring the energy of charged particles produced by neutron-induced nuclear reactions; III. Measuring the flight speed of neutrons (i.e., the time-of-flight method); IV. Determining the minimum energy of a neutron by identifying threshold energy reactions; V. Measuring neutron energy using the principle of neutron diffraction; VI. Determine the energy by measuring the time distribution of the instantaneous slowing down of high-energy neutrons in a specific medium; 7. Based on the detector's overall response to neutron energy, the neutron energy is inferred by deciphering the detector readings.

[0004] In practical applications, neutron energy spectrum measurement is mainly manifested as "neutron energy spectrum inversion", that is, using corresponding mathematical processing methods to solve for a more detailed energy spectrum of the neutron field to be measured based on the measurement values ​​of the detection system.

[0005] Currently, the most widely used inversion method is the least squares method. However, this method essentially solves underdetermined equations, and is extremely sensitive to noise during low-count activation measurements. The spectral results are prone to oscillations or negative fluxes, leading to distortion of the output neutron spectrum.

[0006] Neutron energy spectrum data plays a vital role in fundamental nuclear physics research and neutron application technology research. Developing high-precision and highly stable inversion algorithms can not only directly improve the accuracy of experimental measurements, but also help reduce the stringent requirements on detection system hardware under limited measurement conditions.

[0007] In view of this, the inventors of this application have designed a neutron energy spectrum inversion method based on the maximum likelihood expectation maximization theory in order to overcome the above-mentioned technical problems. Summary of the Invention

[0008] The technical problem to be solved by this invention is to overcome the shortcomings of existing neutron energy spectrum interpretation methods, such as high dependence on prior spectra, poor computational stability, and limited adaptability to various scenarios, and to provide a neutron energy spectrum inversion method based on the maximum likelihood expectation maximization theory.

[0009] The present invention solves the above-mentioned technical problems through the following technical solution: A neutron energy spectrum inversion method based on the maximum likelihood expectation maximization theory is characterized by comprising the following steps: S1. Select at least one set of detector foils; S2. Irradiate the detector foil in the neutron field to be tested, and measure the radioactivity of the detector foil; S3. The neutron energy spectrum is obtained by using the maximum likelihood expectation maximization inversion algorithm.

[0010] According to an embodiment of the present invention, step S1 includes: S 11 The neutron detection system acquires multiple sets of measured data of the neutron field under test through multiple detection units, and converts the acquired detection signals into a counting rate. , where i=1,2,……,n, and n is the number of measurement groups.

[0011] S 12 Obtain the response matrix of the neutron detection system , where j=1,2,……,m, and m is the energy group number; This represents the contribution coefficient of the j-th group of neutrons to the i-th group of measurements.

[0012] According to an embodiment of the present invention, step S2 includes: reading the measured radioactivity of the detector foil. Detection foil activation reaction cross section and prior spectral information .

[0013] According to an embodiment of the present invention, step S3 includes: S 31 Based on the characteristic that the measurement data of the detector unit follows a Poisson distribution, a real neutron energy spectrum is constructed with respect to the detector measurements. The likelihood function satisfies the following relationship:

[0014] in, The likelihood function value represents the current measured data obtained given the true neutron energy spectrum n. The probability of; Represents the vector of the real neutron energy spectrum; This represents the expected count rate of the i-th group of measurements; S 32 Construct the conditional expectation likelihood function: ; in, Let A represent the estimated neutron energy spectrum after the k-th iteration, and A be the measurement value of the detection system. Let C represent the conditional expectation of the sufficient statistics of the neutron energy spectrum after the k-th iteration, where C is a constant.

[0015] S 33 After the k-th iteration The conditional expectation is: ; S 34 Execute the maximum step, for After taking the partial derivative and finding the extreme value, the physically optimal solution that best matches the statistical characteristics of the true neutron energy spectrum is selected from multiple solutions of the underdetermined equation. The iterative update formula for the neutron energy spectrum is then obtained as follows:

[0016] Where i represents the number of detector foils (i=1,2,…,n); j represents the number of energy groups in the neutron energy spectrum to be determined (j=1,2,…,m); Denotes the neutron flux of the j-th group in the k-th iteration; This represents the cross section of the activation reaction of the j-th group of the i-th detector foil; This represents the measured value of the radioactivity of the i-th detector foil.

[0017] According to an embodiment of the present invention, step S 34 This then includes: iteratively optimizing the energy spectrum, with the initial energy spectrum as input. Execute step S 34 The neutron energy spectrum iterative formula introduces smoothing processing during the iteration process.

[0018] According to an embodiment of the present invention, after step S3, step S4 is further included: convergence judgment, calculating the difference in neutron energy spectrum between adjacent iterations, stopping the iteration when the difference is less than a set threshold, and outputting the final energy spectrum. .

[0019] According to an embodiment of the present invention, step S4 includes: setting the iteration convergence limit to 1E-6, when When, output .

[0020] The positive and progressive effects of this invention are as follows: The neutron energy spectrum inversion method based on the maximum likelihood expectation maximization theory has the following advantages: I. The statistical model conforms to the actual neutron detection scenario: Neutron detection counting follows a Poisson distribution. This invention uses the likelihood function based on the Poisson distribution to derive the iterative equation, which is consistent with the statistical characteristics of the detection system, and the inversion results are more in line with physical reality.

[0021] II. Non-physical explanation of natural non-negative flux: The neutron energy spectrum iterative equation of this invention guarantees that the energy spectrum is non-negative, requiring no manual correction and having stronger physical rationality.

[0022] Third, the solution of underdetermined equations is more stable: the present invention does not require direct solution of matrix inverse, and has better resistance to ill-conditioned conditions and higher stability.

[0023] Fourth, it is more robust to noise and statistical fluctuations: Based on stripe Poisson likelihood and conditional expectation, this invention can suppress random fluctuations. Attached Figure Description

[0024] The above and other features, properties and advantages of the present invention will become more apparent from the following description taken in conjunction with the accompanying drawings and embodiments, in which the same reference numerals always denote the same features, wherein: Figure 1 This is a flowchart of the multi-foil activation method in the neutron energy spectrum inversion method based on the maximum likelihood expectation maximization theory of this invention.

[0025] Figure 2 This is a flowchart of the neutron energy spectrum inversion method based on the maximum likelihood expectation maximization theory of the present invention.

[0026] Figure 3 This is a diagram showing the spectral results of the neutron energy spectrum inversion method based on the maximum likelihood expectation maximization theory of this invention. Detailed Implementation

[0027] The present invention will be further described below with reference to specific embodiments and accompanying drawings. More details are set forth in the following description in order to provide a full understanding of the present invention. However, the present invention can obviously be implemented in many other ways different from those described herein. Those skilled in the art can make similar extensions and derivations based on actual application situations without departing from the spirit of the present invention. Therefore, the scope of protection of the present invention should not be limited by the content of this specific embodiment.

[0028] Embodiments of the invention will now be described in detail with reference to the accompanying drawings. It should be noted that these and subsequent drawings are merely illustrative and are not drawn to scale, and should not be construed as limiting the scope of the invention. Wherever possible, the same reference numerals will be used in all drawings to denote the same or similar parts.

[0029] Furthermore, although the terminology used in this invention is selected from commonly known and used terms, some terms mentioned in this specification may have been selected by the applicant in his or her judgment, and their detailed meanings are explained in the relevant sections of the description herein.

[0030] Furthermore, the invention should be understood not only through the actual terminology used, but also through the meaning implied by each term.

[0031] like Figures 1 to 3 As shown, this invention discloses a neutron energy spectrum inversion method based on the maximum likelihood expectation maximization theory. This method involves irradiating a set of detector foils with known activation reaction cross-sections into the neutron field to be measured. After measuring the radioactivity of the detector foils, the neutron energy spectrum is calculated through neutron energy spectrum inversion. It is applicable to energy spectrum inversion of various detection systems, such as the multi-foil activation method, the Bonner sphere spectrometer, and the He-3 counter. The measurement data acquired by the detection system needs to be processed to remove gamma noise, electron noise, and non-neutron background signals.

[0032] The neutron spectrum inversion method based on the maximum likelihood expectation maximization theory includes the following steps: Step S1: Select at least one set of detector foils.

[0033] Initial spectrum selection mechanism: The initial spectrum can be derived from energy spectra under similar conditions in the references, or it can be calculated by deterministic methods or Monte Carlo methods. For neutron energy spectrum inversion, the closer the shape of the initial spectrum is to the true energy spectrum, the higher the calculation accuracy.

[0034] Preferably, step S1 includes: Step S 11 Measured data acquisition: Multiple sets of measured data of the neutron field under test are acquired through the multi-detection unit of the neutron detection system, and the acquired detection signals are converted into count rate. , where i=1,2,……,n, and n is the number of measurement groups.

[0035] Step S 12 Determining basic parameters: Obtaining the response matrix of the neutron detection system. , where j=1,2,……,m, and m is the energy group number; This represents the contribution coefficient of the j-th group of neutrons to the i-th group of measurements.

[0036] Read the measured radioactivity of the detector foil Detector foil activation reaction cross section and prior spectrum information (unit: atom / cm) 2 ).

[0037] Radioactivity processing mechanism: In this embodiment, the input radioactivity of the detector foil must be the mononuclear reactivity of the detector foil. When the radioactivity is not the saturation activity, data processing is required, i.e. ,in To detect the decay constant of the foil.

[0038] Step S2: Irradiate the detector foil in the neutron field to be tested, and measure the radioactivity of the detector foil.

[0039] Here, the activation reaction cross section of the detector foil is known. After measuring the radioactivity of the detector foil, the neutron energy spectrum is obtained by neutron energy spectrum inversion calculation.

[0040] Preferably, step S2 includes: reading the measured radioactivity of the detector foil. (Unit: s) -1 ), detection foil activation reaction cross section (Unit: cm) 2 and prior spectral information (Unit: atom / cm) 2 ).

[0041] Step S3: Use the maximum likelihood expectation maximization inversion algorithm to obtain the neutron energy spectrum.

[0042] Preferably, step S3 includes: Step S 31 Based on the characteristic that the measurement data of the detector unit follows a Poisson distribution, a real neutron energy spectrum is constructed with respect to the detector measurements. The likelihood function satisfies the following relationship:

[0043] in, The likelihood function value represents the current measured data obtained given the true neutron energy spectrum n. The probability of; Represents the vector of the real neutron energy spectrum; This represents the expected count rate for the i-th measurement group.

[0044] Detector measurement value Compared with the real neutron energy spectrum The relationship conforms to the discrete form of the first kind of Fredholm equation. Where i = 1, 2, ..., n is the number of measurement groups; j = 1, 2, ..., m, and m is the number of energy groups. Since m >> n, this system of equations is an underdetermined system of equations, meaning the number of unknowns is greater than the number of independent equations, and the solution to the equations is not unique.

[0045] Constructing the likelihood function: Since the time intervals of neutron incident detection units are independent, the interaction between neutrons and the detection units is recorded as a random event. The measured neutron field is relatively stable within the measurement time period, and the average number of incident neutrons per unit time does not fluctuate significantly. Therefore, the measurement data of the detection units follows a Poisson distribution. Based on this, a likelihood function of the real neutron energy spectrum with respect to the detector measurements is constructed. The likelihood function is the product of the probabilities of each independent measurement group, used to quantify the statistical matching degree between the real neutron energy spectrum and the measured data.

[0046] Step S 32 Construct the conditional expectation likelihood function: ; in, Let A represent the estimated neutron energy spectrum after the k-th iteration, and A be the measurement value of the detection system. Let C represent the conditional expectation of the sufficient statistics of the neutron energy spectrum after the k-th iteration, where C is a constant.

[0047] Since the neutron energy spectrum cannot be measured and determined, it needs to be indirectly reflected through counting by a detection system, rather than directly applied to step S. 31 The likelihood function constructed in the model is difficult to find its extreme value. Therefore, this application introduces the conditional expectation Q to quantify the degree of fit between the neutron energy spectrum and the measurement value of the detection system.

[0048] The conditional expectation likelihood function Q is used to quantify the degree of fit given measured data and the current iterative energy spectrum, transforming the underdetermined equation problem into an iteratively solvable extremum problem and ensuring the monotonic stability of the iterative process. The natural logarithm of the likelihood function is taken when constructing the conditional expectation likelihood function Q because converting the product form into a summation pattern facilitates subsequent partial derivative calculations to obtain extrema, and the optimal energy spectrum obtained after the transformation is completely consistent.

[0049] Step S 33 1. Execute the expected step: After the k-th iteration The conditional expectation is: ; Step S 34 Execute the maximum step, for By taking the partial derivative and finding the extreme value, and selecting the physically optimal solution from multiple solutions of the underdetermined equation that best matches the statistical characteristics of the true neutron energy spectrum, we can obtain the iterative update formula for the neutron energy spectrum, i.e., the neutron flux of the j-th group at the (k+1)-th iteration. for:

[0050] Where i represents the number of detector foils (i=1,2,…,n); j represents the number of energy groups in the neutron energy spectrum to be determined (j=1,2,…,m); Denotes the neutron flux of the j-th group in the k-th iteration; This represents the cross section of the activation reaction of the j-th group of the i-th detector foil; This represents the measured radioactivity value of the i-th detector foil. After each iteration update, the neutron flux is forced to normalize, keeping the total flux rate stable.

[0051] Preferably, step S 34 This then includes: iteratively optimizing the energy spectrum, with the initial energy spectrum as input. Execute step S 34 The neutron energy spectrum iterative formula incorporates a smoothing process during iteration to avoid non-physical singular peaks in the energy spectrum.

[0052] Preferably, step S3 is followed by step S4: convergence judgment, calculating the difference in neutron energy spectrum between adjacent iterations, stopping the iteration when the difference is less than a set threshold, and outputting the final energy spectrum. .

[0053] More preferably, step S4 includes: setting the iteration convergence limit to 1E-6, when When, output .

[0054] Neutron integral energy spectrum (unit: atom / cm) 2 ) and neutron differential energy spectrum (unit: atom / (cm) 2 The results are as follows: (eV) Figure 3 As shown, the calculated neutron energy spectrum exhibits a 1 / v decreasing characteristic in the hot region, fluctuates in the mid-energy region due to resonance peaks, and shows a smooth decay trend in the fast region. It maintains physical continuity and non-negativity across the entire energy range, without numerical oscillations or non-physical spikes. This demonstrates the stability, accuracy, and physical consistency of the algorithm.

[0055] As described above, this invention provides a neutron spectrum inversion method based on the maximum likelihood expectation maximization theory. It calculates high-precision neutron spectrum data by inverting the radioactivity of the activated foil after irradiation, obtained from measurements using the multi-foil activation method. The method addresses the underdetermined equations in the neutron inversion algorithm through iterative cycles of expectation and maximization steps. It constructs a likelihood function based on a Poisson distribution and inversely infers the true neutron spectrum from measured data combined with a physical model. This eliminates the need to directly solve for the inverse non-square matrix, resulting in high inversion accuracy, fast computational efficiency, and compatibility with various detection modes.

[0056] The neutron spectrum inversion method based on the maximum likelihood expectation-maximization theory uses this theory as its core algorithm. The core logic involves iteratively processing underdetermined equations (i.e., the number of measurement groups is less than the number of energy groups, leading to non-unique solutions) through expectation and maximization steps. This allows for indirect deduction of the true neutron spectrum information from measured data combined with a physical model. The theory constructs a likelihood function based on a Poisson distribution, which aligns with the statistical characteristics of neutron detection counting. During iteration, the likelihood function monotonically increases and converges to a stable point, exhibiting low dependence on prior energy spectra. It eliminates the need to directly solve for the inverse of the non-square matrix response matrix, thus avoiding numerical divergence problems. Furthermore, the iteratively updated calculation method is adaptable to processing measurement data from various detection modes, such as pulse and current.

[0057] Compared with the most widely used neutron energy spectrum inversion method, the least squares method, the core advantages of this invention are that the statistical model conforms to the actual neutron detection scenario, naturally satisfies non-negativity, has better robustness to noise and statistical fluctuations, and has better inversion accuracy and stability in low count / complex spectrum scenarios.

[0058] In summary, the neutron energy spectrum inversion method based on the maximum likelihood expectation maximization theory of this invention has the following advantages: I. The statistical model conforms to the actual neutron detection scenario. Neutron detection counting follows a Poisson distribution, while the least squares method is based on the Gaussian distribution assumption. This invention uses a likelihood function based on the Poisson distribution to derive the iterative equation, which is consistent with the statistical characteristics of the detection system, and the inversion results are more in line with physical reality.

[0059] II. Natural Non-Pollution Volume Non-Physical Understanding

[0060] The least squares method is prone to negative energy spectrum solutions under underdetermined conditions and noise perturbations, requiring additional conditional constraints. The neutron energy spectrum iterative equation in this invention guarantees that the energy spectrum is non-negative, requiring no manual correction and having stronger physical rationality.

[0061] III. More stable solution to underdetermined equations

[0062] The least squares method requires direct solution of the matrix inverse or pseudo-inverse, which is prone to divergence and oscillation when the response matrix is ​​ill-conditioned. This invention does not require direct solution of the matrix inverse, and has better resistance to ill-conditioned conditions and higher stability.

[0063] Fourth, it is more robust to noise and statistical fluctuations.

[0064] The least squares method is sensitive to statistical fluctuations in outlier data and is easily misled by local noise. This invention, based on stripe Poisson likelihood and conditional expectation, can suppress random fluctuations.

[0065] For those skilled in the art, the above disclosure is merely illustrative and does not constitute a limitation of this application. Although not explicitly stated herein, those skilled in the art may make various modifications, improvements, and corrections to this application. Such modifications, improvements, and corrections are suggested in this application and therefore remain within the spirit and scope of the exemplary embodiments of this application.

[0066] It should be noted that the use of terms such as "first" and "second" to define components is merely for the purpose of distinguishing the corresponding components. Unless otherwise stated, the above terms have no special meaning and therefore should not be construed as limiting the scope of protection of this application.

[0067] In the description of this application, it should be understood that the orientation or positional relationship indicated by directional terms such as "front, back, up, down, left, right", "horizontal, vertical, horizontal" and "top, bottom" is usually based on the orientation or positional relationship shown in the accompanying drawings, and is only for the convenience of describing this application and simplifying the description. Unless otherwise stated, these directional terms do not indicate or imply that the device or element referred to must have a specific orientation or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation on the scope of protection of this application; the directional terms "inner" and "outer" refer to the inner and outer contours relative to the outline of each component itself.

[0068] For ease of description, spatial relative terms such as "above," "on top of," "on the upper surface of," "above," etc., are used herein to describe the spatial positional relationship of a device or feature as shown in the figures to other devices or features. It should be understood that spatial relative terms are intended to encompass different orientations in use or operation beyond the orientation of the device as described in the figures. For example, if the device in the figures were inverted, a device described as "above" or "on top of" other devices or structures would subsequently be positioned as "below" or "under" other devices or structures. Thus, the exemplary term "above" can include both "above" and "below." The device may also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatial relative descriptions used herein will be interpreted accordingly.

[0069] Furthermore, this application uses specific terms to describe embodiments of the application. For example, "an embodiment," "one embodiment," and / or "some embodiments" refer to a particular feature, structure, or characteristic related to at least one embodiment of the application. Therefore, it should be emphasized and noted that "an embodiment," "one embodiment," or "an alternative embodiment" mentioned twice or more in different locations in this specification do not necessarily refer to the same embodiment. In addition, certain features, structures, or characteristics in one or more embodiments of the application can be appropriately combined.

[0070] Similarly, it should be noted that, in order to simplify the description of the embodiments disclosed in this application and thus aid in the understanding of one or more embodiments of the invention, the foregoing description of the embodiments of this application sometimes combines multiple features into one embodiment, drawing, or description thereof. However, this disclosure method does not mean that the subject matter of this application requires more features than those mentioned in the claims. In fact, the embodiments have fewer features than all the features of the single embodiments disclosed above. Some embodiments use numbers describing the quantity of components and attributes. It should be understood that such numbers used in the description of embodiments are modified in some examples by the terms "approximately," "approximately," or "generally." Unless otherwise stated, "approximately," "approximately," or "generally" indicates that the numbers are allowed to vary by ±20%.

[0071] Accordingly, in some embodiments, the numerical parameters used in the specification and claims are approximate values, which may be changed depending on the characteristics required by individual embodiments. In some embodiments, the numerical parameters should take into account specified significant digits and employ a general method of digit preservation. Although the numerical ranges and parameters used to confirm their breadth of application in some embodiments of this application are approximate values, in specific embodiments, such values ​​are set as precisely as feasible.

[0072] While specific embodiments of the present invention have been described above, those skilled in the art should understand that these are merely illustrative examples, and the scope of protection of the present invention is defined by the appended claims. Those skilled in the art can make various changes or modifications to these embodiments without departing from the principles and essence of the present invention, but all such changes and modifications fall within the scope of protection of the present invention.

Claims

1. A neutron energy spectrum inversion method based on the maximum likelihood expectation maximization theory, characterized in that, The neutron energy spectrum inversion method includes the following steps: S1. Select at least one set of detector foils; S2. Irradiate the detector foil in the neutron field to be tested, and measure the radioactivity of the detector foil; S3. The neutron energy spectrum is obtained by using the maximum likelihood expectation maximization inversion algorithm.

2. The neutron spectrum inversion method based on the maximum likelihood expectation maximization theory as described in claim 1, characterized in that, Step S1 includes: S 11 The neutron detection system uses multiple detection units to collect multiple sets of measured data of the neutron field under test, and converts the collected detection signals into a counting rate. , where i=1,2,……,n, and n is the number of measurement groups. S 12 Obtain the response matrix of the neutron detection system , where j=1,2,……,m, and m is the energy group number; This represents the contribution coefficient of the j-th group of neutrons to the i-th group of measurements.

3. The neutron spectrum inversion method based on the maximum likelihood expectation maximization theory as described in claim 1, characterized in that, Step S2 includes: reading the measured radioactivity of the detector foil. Detection foil activation reaction cross section and prior spectrum information .

4. The neutron spectrum inversion method based on the maximum likelihood expectation maximization theory as described in claim 1, characterized in that, Step S3 includes: S 31 Based on the characteristic that the measurement data of the detector unit follows a Poisson distribution, a real neutron energy spectrum is constructed with respect to the detector measurements. The likelihood function satisfies the following relationship: in, The likelihood function value represents the current measured data obtained given the true neutron energy spectrum n. The probability of; Represents the vector of the real neutron energy spectrum; This represents the expected count rate of the i-th group of measurements; S 32 Construct the conditional expectation likelihood function: ； in, Let A represent the estimated neutron energy spectrum after the k-th iteration, and A be the measurement value of the detection system. Let C represent the conditional expectation of the sufficient statistics of the neutron energy spectrum after the k-th iteration, where C is a constant. S 33 After the k-th iteration The conditional expectation is: ； S 34 Execute the maximum step, for After taking the partial derivative and finding the extreme value, the physically optimal solution that best matches the statistical characteristics of the true neutron energy spectrum is selected from multiple solutions of the underdetermined equation. The iterative update formula for the neutron energy spectrum is then obtained as follows: Where i represents the number of detector foils (i=1,2,…,n); j represents the number of energy groups in the neutron energy spectrum to be determined (j=1,2,…,m); Denotes the neutron flux of the j-th group in the k-th iteration; This represents the cross section of the activation reaction of the j-th group of the i-th detector foil; This represents the measured value of the radioactivity of the i-th detector foil.

5. The neutron spectrum inversion method based on the maximum likelihood expectation maximization theory as described in claim 1, characterized in that, The step S 34 This then includes: iteratively optimizing the energy spectrum, with the initial energy spectrum as input. Execute step S 34 The neutron energy spectrum iterative formula introduces smoothing processing during the iteration process.

6. The neutron spectrum inversion method based on the maximum likelihood expectation maximization theory as described in claim 1, characterized in that, Following step S3, step S4 is also included: convergence judgment, calculating the difference in neutron energy spectrum between adjacent iterations, stopping the iteration when the difference is less than a set threshold, and outputting the final energy spectrum. .

7. The neutron spectrum inversion method based on the maximum likelihood expectation maximization theory as described in claim 6, characterized in that, Step S4 includes: setting the iteration convergence limit to 1E-6, when When, output .

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